Center of Mass

The center of mass is the center of a system of masses, located at a point that is central in relation to the masses of all of the objects within the system.  To describe the position of the center of mass, we use the equation$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{\sum\limits_{i=0}^n m_i}$$$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{M_{total}}$$

The velocity of the center of mass is the same thing, except finding velocity instead of position.$$\vec{v}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{v}_i}{M_{total}}$$

If the velocity is very small compared to the speed of light, $\gamma = 1$ we can say that the total velocities and masses of the system are equal based off of the equation for momentum.  This helps us find the momentum of the center of mass:$$\vec{p} = \gamma\times m\vec{v}$$$$\vec{p}_{sys} = M_{total}\vec{v}_{CM}$$$$\vec{p}_{sys} = \sum\limits_{i=0}^n m_i \vec{v}_i$$

Momentum

In real life, it seems obvious that stopping a baseball traveling at high speed is much easier than, say, a refrigerator hurdling towards you at high speed.  The mass, or weight, of the refrigerator is much greater than the baseball!  Everyone knows that it becomes increasingly difficult to move or change something as it becomes heavier, but how are we to describe this phenomenon using physics?

In order to represent the combination of an object's mass ($m$) and velocity ($\vec{v}$), a vector quantity called momentum, $\vec{p}$, is defined.  Notice that mass is a scalar and velocity a vector.  In our visible, daily lives, this equation exists simply as $\vec{p} = m \times \vec{v}$.  However there is another important factor that we must account for.

Certain experiments have shown that as particles travel closer and closer to the speed of light ($c = 3 \times 10^8$ m/s), the amount of interaction required in order to create an increase in velocity becomes increasingly large.  This is Einstein's "relativistic" definition of momentum, defined below.

In order to model the disparity between low speed velocity and velocity that approaches the speed of light, we introduce the scalar proportionality factor, gamma $\gamma$, which is equal to:  $$\gamma = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}}$$
Using this proportionality factor, we can now write the correct equation to model momentum at all speeds:  $$\vec{p} = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}} m \vec{v}$$ $$\vec{p} = \gamma \times m \vec{v}$$

Generally, gamma is approximately exactly 1 at low speeds, and is only effective at very high speeds.  We can simplify our momentum equations in low speed situations for the sake of ease down to:  $$\vec{p} = 1 \times m \vec{v} = m \vec{v}$$

Momentum is the integral of the net force ($\vec{F}_{net}$)$$\vec{p} = \int \vec{F}_{net}$$Using the definition of acceleration and its relation to velocity ($\vec{a} = \frac{\delta \vec{v}}{\delta t}$), we can define the approximate rate of change of momentum as$$\frac{\delta \vec{p}}{\delta t} = m\vec{a}$$$$\vec{F}_{net} = (\frac{\delta \vec{p}}{\delta t} = m\frac{\delta \vec{v}}{\delta t}) = m\vec{a}$$

Acceleration

Acceleration is the time rate of change of velocity, just as velocity is the time rate of change of position.

Instantaneous acceleration can be calculated as the time rate of change (derivative) of velocity:  $$\vec{a} = \frac{\delta \vec{v}}{\delta t}$$
Average acceleration can be calculated as a change in velocity:  $$\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$$

Position Update Formula

The position update formula simply states that given the initial position ($\vec{r}_i$) of an object, we can predict the final position ($\vec{r}_f$) using the average velocity ($\vec{v}_{avg}$) and change in time ($\Delta t$).

$$\Delta \vec{r} = \vec{v}_{avg} \Delta t$$
$$\vec{r}_f - \vec{r}_i = \vec{v}_{avg}(t_f - t_i)$$

Or just simply

$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t$$

Instantaneous Velocity

Instantaneous velocity measures the precise velocity of the ball at a given position.  We can approximate the instantaneous velocity by using the average velocity ($\vec{v}_{avg}$) from one position to another ($\Delta \vec{r}$) over a time interval ($\Delta t$), but it will be unlikely to be accurate.  In order to increase the accuracy of the measurement of the instantaneous velocity, we must make the time interval and position interval increasingly small.  The smaller the position and time interval, the more precise a value we will obtain for the instantaneous velocity:
$$\vec{v} = \lim_{\Delta t \to 0}\frac{\Delta \vec{r}}{\Delta t}$$

Otherwise known in calculus as

$$\vec{v} = \frac{\delta \vec{r}}{\delta t}$$

 As the values become smaller, they become more immediate, or "instantaneous."  The more complicated explanation of this equation:

$$\vec{v} = \frac{\delta\vec{r}}{\delta t} = \frac{\delta}{\delta t}<x, y, z> = <\frac{\delta x}{\delta t}, \frac{\delta y}{\delta t}, \frac{\delta z}{\delta t}> = <v_x, v_y, v_z>$$

This gives us the definition of velocity as being the "time rate of change of position:  $\vec{v} = \frac{\delta \vec{r}}{\delta t}$."

Average Velocity

Velocity is uses to describe the speed and direction of an object.  If we know the object's speed at a certain time as well as the direction it faces, then we can predict where it will be in the future.

A common way to determine the velocity of an object is to find its average velocity ($\vec{v}_{avg}$) - the distance ($\Delta \vec{r}$) it has traveled divided by the elapsed time ($\Delta t$) it took to travel.

$$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$$

This equation can be rearranged to determine the distance traveled, $\Delta \vec{r} = \vec{v}_{avg}\times \Delta t$, and time elapsed, $\Delta t = \frac{\Delta \vec{r}}{\vec{v}_{avg}}$.

In a situation where the rate of change of velocity is not constant, we instead use the equation:$$\vec{v}_{avg} = \frac{\sum\limits_{i=1}^n \vec{v}_i}{n}$$